
Mark Hoefer's Research and AnimationsResearch in physical applied mathematics, motivated by real world problems is central to my investigations. My research is generally in the field of nonlinear waves with a focus on two areas: 1) fluid dynamics of dispersive media with accompanying wave excitations including dispersive shock waves (DSWs) and solitary waves; 2) dynamics of ferromagnetic media, spin torque, and localized excitations in nanomagnetism. Methods employed include mathematical modeling, analysis, asymptotics, Whitham modulation theory, and numerical analysis. Whenever possible, comparisons with experiment are carried out.The generation of DSWs represents a universal mechanism to resolve hydrodynamic singularities in dispersive media. Physical manifestations of DSWs include undular bores on shallow water and in the atmosphere (the Morning Glory), nonlinear diffraction patterns in optics, and matter waves in ultracold atoms. Any approximately conservative, nonlinear, hydrodynamic medium exhibiting weak dispersion can develop DSWs. The mathematical description of DSWs involves a synthesis of methods from hyperbolic quasilinear systems, asymptotics, and soliton theory. This research is supported by the National Science Foundation through grants DMS1008973 and CAREER DMS1255422. Ferromagnetic media provide a source of rich nonlinear, dispersive phenomena with practical import. Theoretical and technological developments have stimulated the field of nanomagnetism by the introduction of spin polarized currents as a means to excite magnetization dynamics at the nanometer scale in patterned environments. Strongly nonlinear magnetic solitons were recently observed in a nanomagnetic system. This solitary wave or "droplet" joins the domain wall and magnetic vortex as a fundamental and distinct object in nanomagnetism with similar potential for fruitful science. This research is supported by the National Science Foundation through a Career award, DMS1255422. A dynamic, visual representation of some of my scientific work is exhibited in the linked pages below with animations of numerical simulations from some problems that I have considered. Numerical simulations are excellent for describing a particular problem with a particular set of parameters. However, to understand the qualitative behavior, and to be able to make informed predictions, mathematical analysis is essential. The mathematical discussion has been kept to a minimum. If you're interested in the math, read the papers! Not all of my publications are represented here. To see a list, click here. The animations can be viewed with Quicktime. Note that some of the simulation files can be somewhat large.
